The constant-term formula for the $(1,2)$ double-torus curve

Let $N$ be a positive integer, let $q_u=q^{\frac{1}{2}}$, and let $S_{N-1}(\mathcal{A}_{(1,2)})$ be the indicated Chebyshev-type expression in the DAHA operator $\mathcal{A}_{(1,2)}$. Write $\operatorname{Const}$ for the constant term $\eth^0$ and let $\mathbf{v}_N$, $\mathbf{B}_N$, and $\mathbf{T}_N$ be the vectors and matrices defined in the surrounding construction; in particular, $\mathbf{T}_N$ is diagonal with entries indexed by $1\leq k\leq N$. Constant-term formula. The constant term at $t=q_u=q^{\frac{1}{2}}$ is given by

$$\operatorname{Const}\left(\left.S_{N-1}(\mathcal{A}_{(1,2)})\right|_{t=q_u}\right)=(-1)^{N-1}q^{\frac{1}{2}(N-1)}\frac{1-q}{1-q^N}\,\mathbf{v}_N(q,\mathsf{X}_u)\,\mathbf{T}_N(q;\mathbb{c}_{(1,2)})\,\mathbf{v}_N(q,\mathsf{X}_d)^\top.$$

Here $\mathbf{T}_N$ has diagonal entries $(-1)^{k-1}q^{\frac{1}{2}k(k+1-2N)}\frac{(q^{2k};q)_{N+1-2k}}{(q^k;q)_{N+1-2k}}\frac{(q^{2k};q)_{N-k}}{(q^k;q)_{N-k}}$. The formula is presented as an observation based on explicit computations and is used to calculate reduced DAHA polynomials; the supplied text gives no proof or resolution.